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Chapter 2: Problem 83

A savings account earns \(5 \%\) interest compounded annually. If you invest \(x\) dollars in such an account, then the amount \(A(x)\) of the investment after one year is the initial investment plus \(5 \% ;\) that is, $$A(x)=x+0.05 x=1.05 x$$ Find $$\begin{aligned} &A \circ A\\\&A \circ A \circ A\\\&A \circ A \circ A \circ A\end{aligned}$$ What do these compositions represent? Find a formula for what you get when you compose \(n\) copies of \(A\).

Short Answer

Expert verified
\(A^n(x) = (1.05)^n x\), representing the amount after \(n\) years with \(5\%\) interest.

Step by step solution

01

Understand the Function

The function \(A(x) = 1.05x\) represents the amount of money in the savings account after one year, starting with \(x\) dollars and considering \(5\%\) annual interest.
02

Compute A(A(x))

To find \(A \circ A\), substitute \(A(x)\) into itself: \[A(A(x)) = A(1.05x) = 1.05(1.05x) = (1.05)^2x\]. This shows the amount after two years of interest.
03

Compute A(A(A(x)))

Find \(A \circ A \circ A\) by applying \(A\) to \(A(A(x))\): \[A(A(A(x))) = A((1.05)^2x) = 1.05((1.05)^2x) = (1.05)^3x\]. This is the amount after three years of compounding.
04

Compute A(A(A(A(x))))

Compute \(A \circ A \circ A \circ A\) by substituting into the previous result: \[A(A(A(A(x)))) = A((1.05)^3x) = 1.05((1.05)^3x) = (1.05)^4x\]. This is the amount after four years.
05

Generalize the composition for n years

Notice the pattern from the previous steps: \(A^n(x) = (1.05)^n x\). This formula represents the amount of money in the account after \(n\) years of compounding.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Investment Growth
Investment growth is about how your money increases over time when it's invested. Let's think about planting a seed. When you plant a seed, it grows into a plant. Similarly, when you "plant" money in a savings account, it grows due to interest.
  • Initially, you have your principal, which is the original amount of money you start with.
  • With compound interest, your investment grows as it earns interest on both the initial principal and on the already-earned interest.
The idea is to let your investment grow by keeping it in the account over time. The longer it stays, the more interest it earns, like letting the plant keep growing in the sun. Each year, the growth is based on the previous year's amount, not just the initial amount. So, the more patient you are, the larger the return will be.
Composition of Functions
The composition of functions is like applying one rule after another. Imagine getting dressed; you first put on a shirt, then a sweater over it. Each layer you add changes the overall look.
  • Here, we apply the function \(A(x)\) repeatedly to itself, which means we're reapplying the rule for adding interest, again and again.
  • Each function we apply represents another year passing with compound interest in action.
When we write \(A \circ A \), it's like saying, "What happens if I apply the interest rule on my already increased amount?" By the time we do \(A \circ A \circ A\), we've completed three years of interest application.The general formula \( (1.05)^n x \) shows how your repeated applications stack up over \( n \) years, just like adding more and more layers to your outfit.
Annual Interest Rate
The annual interest rate is like the pace at which your savings grow each year. If you imagine a runner, the interest rate is the speed at which the runner is moving.
  • Let's say an annual interest rate is \(5\%\), which means for every dollar in your account, you gain an extra \(5\%\) each year.
  • This is particularly important when it's compounded annually because it means the interest for next year will not only include the original amount, but also what was gained in interest last year.
Essentially, a high annual interest rate means your money grows faster each year. The idea is similar to compounding force in physics 鈥� it builds upon itself every cycle, making your investment larger and more robust over time. In our example, a consistent \(5\%\) rate leads to exponential growth, as seen in the compounded formula \( 1.05^n x \). Letting it accumulate without interruption is how you maximize growth.

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Most popular questions from this chapter

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(h(x)=x^{3}-5 x-4\) (a) \([-2,2]\) by \([-2,2]\) (b) \([-3,3]\) by \([-10,10]\) (c) \([-3,3]\) by \([-10,5]\) (d) \([-10,10]\) by \([-10,10]\)

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of \(60 \mathrm{cm} / \mathrm{s}\). (a) Find a function \(g\) that models the radius as a function of time. (b) Find a function \(f\) that models the area of the circle as a function of the radius. (c) Find \(f \circ g .\) What does this function represent?

An object is dropped from a high cliff, and the distance (in feet) it has fallen after \(t\) seconds is given by the function \(d(t)=16 t^{2}\) Complete the table to find the average speed during the given time intervals. Use the table to determine what value the average speed approaches as the time intervals get smaller and smaller. Is it reasonable to say that this value is the speed of the object at the instant \(t=3 ?\) Explain. $$\begin{array}{|c|c|c|} \hline t=a & t=b & \text { Average speed }=\frac{d(b)-d(a)}{b-a} \\ \hline 3 & 3.5 & \\ 3 & 3.1 & \\ 3 & 3.01 & \\ 3 & 3.001 & \\ 3 & 3.0001 & \\ \hline \end{array}$$

The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 0.9766 U.S. dollars. (a) Find a function \(f\) that gives the U.S. dollar value \(f(x)\) of x Canadian dollars. (b) Find \(f^{-1}\). What does \(f^{-1}\) represent? (c) How much Canadian money would \(\$ 12,250\) in U.S. currency be worth?

These exercises show how the graph of \(y=|f(x)|\) is obtained from the graph of \(y=f(x)\) (Graph can't copy) The graphs of \(f(x)=x^{2}-4\) and \(g(x)=\left|x^{2}-4\right|\) are shown. Explain how the graph of \(g\) is obtained from the graph of \(f\)
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